Rank-r latent models for cross-covariance∗
نویسندگان
چکیده
We specify a class of Gaussian rank-r latent models for cross-covariance. We show by construction that any variance-covariance matrix for the observed variables induced by rank-r reduced-rank regression can be induced by a rank-r latent model. 1 Model specification Basic terms are introduced which will be used to state the result. 1.1 Rank-r constraint models Let p be the number of X-variables and q the number of Y-variables. The rank-r symmetric constraint model (equivalently, the rank-r reduced-rankregression model) is the set of (p + q) × (p + q) positive semidefinite matrices satisfying a rank constraint on the cross-covariance matrix: Σ = [ ΣXX ΣXY ΣY X ΣY Y ] , where ΣXY is p× q of rank r. (1) ∗Technical Report No. 411, University of Washington, Department of Statistics, Box 354322, Seattle WA 98195, U.S.A. †Research supported in part by the U.S. EPA and by National Science Foundation Grants No. DMS-0071818 and No. DMS-9972008. The author thanks David Ragozin of the University of Washington Department of Mathematics, who gave generously of his time in several provocative and helpful conversations.
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